1. ## vector problem

A rectangle PQRS where point A is on PQ such that PA = 2/3 PQ . AS intersects PR at B . It is given that SP = 4 q , SR = 6p , |p| = 3 units and |q| = 2 units . Find vector PB and BA .

2. I just noticed now that these are supposed to solved using vectors.

I know vectors. The old way. Even if I studied the new or modern way in dealing with vectors, my Math brain rejected this new way.
Just like using matrices in solving simultaneous equations, my stubborn Math brain rejected the matrices.
(Those two ways are making easy Math hard.)

So, I will show a solution not using the vector way. It's just through simple Geometry, and/or trig.

In triangle PBA:

Referring to the right triangle PQR,
tanP = QR /PQ = 4(2) /6(3) = 8/18
angle P = arctan(8/18) = 23.9625 degrees

Referring to right triangle APS,
tanA = 8 / (2/3)18 = 8/12
angle A = arctan(8/12) = 33.69 deg

So, angle B = 180 -23.9625 -33.69 = 122.3475 deg

By Law of Sines,
AP /sin(122.3475deg) = PB /sin(33.69deg)
12 /sin(122.3475deg) = PB /sin(33.69deg)